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Myerson: Bayesian Single-item Auction problem: ◦ Single item for sale (can be extended to

``service”)◦ bidders◦ Distribution from which bidder values are drawn◦ VCG on virtual values

Recall Myerson Optimal Auction

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What about Myerson when bidders are correlated? (Not a product distribution)

Answer: Can beat Myerson revenue (by cheating somewhat)

Exp. Utility to agent if value = 10 is zero Exp. Utility to agent if value = 100 is 30

Quick Question (to make sure everyone is awake):

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Run VCG AND charge extra payments: P(b) – extra payment if OTHER GUY bids b

Both bid 100: both get utility -60 Both bid 10: both get utility +30

Joint distribution

Revenue = TOTAL “surplus”Better than Reserve price of 100

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On Profit-Maximizing Envy-Free Pricing◦ Guruswami, Hartline, Karlin, Kempe, Kenyon, McSherry◦ SODA 05

Algorithmic Pricing via Virtual Valuations◦ Chawla, Hartline, Kleinberg◦ arXiv 2008

Pricing Randomized Allocations◦ Briest, Chawla, Kleinberg, Weinberg ◦ arXiv, 2009

Approximate Revenue Maximization with Multiple Items◦ Hart, Nisan◦ arXiv 2012

Papers today (very partially)

Myerson: Bayesian Single-item Auction problem: ◦ Single item for sale (can be extended to ``service”)◦ bidders◦ Distribution from which bidder values are drawn◦ VCG on virtual values

Bayesian Unit-demand Pricing Problem: ◦ Single unit-demand bidder◦ items for sale◦ Distribution from which the bidder valuation for

items is drawn

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Bayesian Unit Demand Pricing Problem

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How do n bidders single item relate to single bidder n items?

What’s the connection

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For the Bayesian Single item Auction and the Bayesian Unit-demand Pricing problem are the same problem.

Offer the item at a price of where is the virtual valuation function for distribution

𝒏=𝟏

Ái (vi ) = vi ¡1¡ F i (vi )

f i (vi ): (For regular distributions,

Use ironed virtual values if irregular)

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Theorem: For any price vector p, the revenue of Myerson (when bidder value of the single item comes from is the revenue when the single bidder gets value for item from

Given price vector p, new mechanism M: ◦ Allocate the item to the bidder with that maximizes ◦ Have bidder pay the critical price (the lowest bid at

which bidder would win. The allocation function is monotone and so

this is a truthful mechanism. Myerson is optimal amongst all Bayes Nash IC

mechanisms, ◦ Myerson gets more revenue than M

𝒏≥𝟏

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◦ Allocate the item to the bidder with that maximizes ◦ Have bidder pay the critical price (the lowest bid at which

bidder would win.

The minimum bid for bidder to win is This is the revenue to M when wins

The revenue from a single bidder who buys item with value at a price of is

QED

Myerson Revenue is Opt revenue for Single bidder Bayesian values

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Let - the probability that no item is sold at pricing vector

Use prices , where is chosen as follows: ◦ Choose such that and

Theorem (Chawla, Hartline, B. Kleinberg): Gives no less than Myerson revenue / 3◦ Algorithmic Pricing via Virtual Valuations

Reg distributions: almost tight (??)

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Discrete Distributions

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The following two problems are equivalent: ◦ Given n items and m “value vectors” for m

bidders compute the revenue-optimal item pricing

◦ Given a single bidder whose valuations are chosen uniformly at random from compute the revenue-optimal item pricing

Distribution vs. set of Bidders

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Reduction from vertex cover on graphs of maximum degree at most B (APX hard for B3).◦ Given (connected) Graph G with n vertices

construct n items and m+n bidders (or bidder types)

◦ For every edge add bidder with value 1 for items and zero otherwise

◦ Additionally, for every item there is a bidder with value 2 for the item.

◦ The optimal pricing gives profit, where is the smallest vertex cover of

APX hardness of optimal (deterministic) item pricing

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◦ Given (connected) Graph G with n vertices construct n items and m+n bidders (or bidder types)

◦ For every edge add bidder with value 1 for items and zero otherwise◦ Additionally, for every item there is a bidder with value 2 for the item. ◦ The optimal pricing gives profit, where is the smallest vertex cover of

Let S be vertex cover, charge 1 for items in S, 2 otherwise

If and price(i)=price(j)=2 then no profit from bidder - reducing price of (say) to one keeps profit unchange.

APX hardness of (deterministic) item pricing

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Pricing Randomized Allocations◦ Patrick Briest, Shuchi Chawla, Robert Kleinberg, S.

Matthew Weinberg◦ Remark: When speaking, say “Bobby Kleinberg”,

when writing write “Robert Kleinberg”.

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2 items uniformly value uniformly distributed in [a,b]

Optimal item pricing sets

Offer lottery equal prob item 1 or item 2, at cost

Lose here Gain here

Selling Lotteries

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Buy-one model: Consumer can only buy one option

Buy-many model: Consumer can buy any number of lotteries and get independent sample from each (will discard multiple copies)

Different models

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Polytime algorithm to compute optimal lottery pricing

For item pricing (also known as envy-free unit demand pricing) the optimal item pricing is APX-hard (earlier today)

There is no finite ratio between optimal lottery revenue and optimal item-pricing revenue (for

Buy-one model

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where , and price of lottery – collection of lotteries Bidder type is Utility of picking lottery is

Utility maximizing lotteries Max payment for utility maximizing lottery:

Profit of is

Lotteries

u(v;¸) = (P n

i=1 Ái vi ) ¡ p

p+(v;¤) = maxfpj(Á;p) 2 ¤(v)g:

¼(¤) =Z

p+(v;¤)dD

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Poly time optimal algorithm in case of finite support

values of type j bidders - prob of type j bidders

– lottery designed for type j - price for lottery

𝑥 𝑗𝑖≥0 ,𝑧 𝑗≥0

Feasible

Affordable

Type j prefer lottery j

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- maximal expected revenue for single bidder when offered optimal revenue maximizing lotteries, bidder values from (joint) distribution

- maximal expected revenue for single bidder when offered optimal revenue maximizing item prices, bidder values from (joint) distribution

Compare lottery revenue with item pricing revenue

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Theorem:

𝒏=𝟏¤ = f (Á0;p0);(Á1;p1); : : : ; (Ám;pm)g;

0 = Á0 < Á1 < ¢¢¢< Ám;

0 = p0 < p1 < ¢¢¢< pm:

Assume optimal

WLOG

Valuation chooses to purchase

Áj v ¡ pj ¸ Áj ¡ 1v ¡ pj +1

Áj v ¡ pj ¸ Áj +1v¡ pj +1

v 2·

pj ¡ pj ¡ 1

Áj ¡ Áj ¡ 1;

pj +1 ¡ pj

Áj +1 ¡ Áj

¸

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Valuation chooses to purchase

Áj v ¡ pj ¸ Áj ¡ 1v ¡ pj +1

Áj v ¡ pj ¸ Áj +1v¡ pj +1

v 2·

pj ¡ pj ¡ 1

Áj ¡ Áj ¡ 1;

pj +1 ¡ pj

Áj +1 ¡ Áj

¸

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Correlated distributions, unit demand, finite distribution over types: ◦ Deterministic, item pricing

We saw: APX hardness Lower bound of (m- number of items) or - number of

different agent types – not standard complexity assumptions Briest

◦ We saw: Lotteries: optimal solvable Briest et al Product distributions:

◦ 2 approximation Chawla Hatline Malec Sivan◦ PTAS – Cai and Daskalakis

Making sense of it (again)

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bidders, 1 item 1 bidder, n items

2 Approximation product distribution

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Gambler◦ games◦ Each game has a payoff drawn from an independent

distribution◦ After seeing payoff of i th game, can continue to

next game or take payoff and leave◦ Each payoff comes from a distribution◦ Optimal gambler stategy: backwards induction

Theorem: There is a threshold t such that if the gambler takes the first prize that exceeds t then the gambler gets ½ of the maximal payoff

Prophet Inequality

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Proof

Benchmark (highest possible profit for gambler):

For

Choose such that (exists?) nY

i=1

F i (t) = 1=2:

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Profit of gambler = “extras” - event that for all :

Prob(²i ) =Y

j 6=i

F j (t) ¸Y

j

F j (t) = 1=2:

E [max(vi ¡ t;0)j²i ] = E [max(vi ¡ t;0)]

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Proof of Prophet InequalityBenchmark

Gambler profit

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More generally

Find t such that increases and is continuousdecreases and is continuous

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Many bidders, one item, valuation for bidder from ◦ Vickrey 2nd price auction maximzes social welfare

(◦ Myerson maximize virtual social welfare

maximizes revenue Many items, one bidder, value for item

from

Relationship to Auctions

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Many bidders, 1 item: ◦ There is an anonymous (identical) posted price

that gives a approximation to welfare. Directly from Prophet inequality.

◦ Who gets the item? Could be anyone that wants it. There is no restriction on tie breaking.

Many items, one bidder: ◦ There is a single posted price (for all items) that

gives a approximation to social welfare.

Simpler social welfare maximization

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Many bidders, 1 item: ◦ There is a single virtual price that gives a

approximation to welfare. Directly from Prophet inequality and that the revenue is the expected virtual welfare.

◦ This means non-anonymous prices ◦ Who gets the item? Could be anyone that wants

it. There is no restriction on tie breaking. Many items, one bidder:

◦ Use same prices for approximation

Simpler revenue maximization

Á¡ 1i (t)